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1314. 矩阵区域和

题目描述

给你一个 m x n 的矩阵 mat 和一个整数 k ，请你返回一个矩阵 answer ，其中每个 answer[i][j] 是所有满足下述条件的元素 mat[r][c] 的和： 

  • i - k <= r <= i + k,
  • j - k <= c <= j + k 且
  • (r, c) 在矩阵内。

 

示例 1：

输入：mat = [[1,2,3],[4,5,6],[7,8,9]], k = 1
输出：[[12,21,16],[27,45,33],[24,39,28]]

示例 2：

输入：mat = [[1,2,3],[4,5,6],[7,8,9]], k = 2
输出：[[45,45,45],[45,45,45],[45,45,45]]

 

提示：

  • m == mat.length
  • n == mat[i].length
  • 1 <= m, n, k <= 100
  • 1 <= mat[i][j] <= 100

解法

方法一：二维前缀和

本题属于二维前缀和模板题。

我们定义 $s[i][j]$ 表示矩阵 $mat$ 前 $i$ 行，前 $j$ 列的元素和。那么 $s[i][j]$ 的计算公式为：

$$ s[i][j] = s[i-1][j] + s[i][j-1] - s[i-1][j-1] + mat[i-1][j-1] $$

这样我们就可以通过 $s$ 数组快速计算出任意矩形区域的元素和。

对于一个左上角坐标为 $(x_1, y_1)$，右下角坐标为 $(x_2, y_2)$ 的矩形区域的元素和，我们可以通过 $s$ 数组计算出来：

$$ s[x_2+1][y_2+1] - s[x_1][y_2+1] - s[x_2+1][y_1] + s[x_1][y_1] $$

时间复杂度 $O(m \times n)$，空间复杂度 $O(m \times n)$。其中 $m$ 和 $n$ 分别是矩阵的行数和列数。

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class Solution:
    def matrixBlockSum(self, mat: List[List[int]], k: int) -> List[List[int]]:
        m, n = len(mat), len(mat[0])
        s = [[0] * (n + 1) for _ in range(m + 1)]
        for i, row in enumerate(mat, 1):
            for j, x in enumerate(row, 1):
                s[i][j] = s[i - 1][j] + s[i][j - 1] - s[i - 1][j - 1] + x
        ans = [[0] * n for _ in range(m)]
        for i in range(m):
            for j in range(n):
                x1, y1 = max(i - k, 0), max(j - k, 0)
                x2, y2 = min(m - 1, i + k), min(n - 1, j + k)
                ans[i][j] = (
                    s[x2 + 1][y2 + 1] - s[x1][y2 + 1] - s[x2 + 1][y1] + s[x1][y1]
                )
        return ans

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class Solution {
    public int[][] matrixBlockSum(int[][] mat, int k) {
        int m = mat.length;
        int n = mat[0].length;
        int[][] s = new int[m + 1][n + 1];
        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < n; ++j) {
                s[i + 1][j + 1] = s[i][j + 1] + s[i + 1][j] - s[i][j] + mat[i][j];
            }
        }

        int[][] ans = new int[m][n];
        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < n; ++j) {
                int x1 = Math.max(i - k, 0);
                int y1 = Math.max(j - k, 0);
                int x2 = Math.min(m - 1, i + k);
                int y2 = Math.min(n - 1, j + k);
                ans[i][j] = s[x2 + 1][y2 + 1] - s[x1][y2 + 1] - s[x2 + 1][y1] + s[x1][y1];
            }
        }
        return ans;
    }
}

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class Solution {
public:
    vector<vector<int>> matrixBlockSum(vector<vector<int>>& mat, int k) {
        int m = mat.size();
        int n = mat[0].size();

        vector<vector<int>> s(m + 1, vector<int>(n + 1));
        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < n; ++j) {
                s[i + 1][j + 1] = s[i][j + 1] + s[i + 1][j] - s[i][j] + mat[i][j];
            }
        }

        vector<vector<int>> ans(m, vector<int>(n));
        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < n; ++j) {
                int x1 = max(i - k, 0);
                int y1 = max(j - k, 0);
                int x2 = min(m - 1, i + k);
                int y2 = min(n - 1, j + k);
                ans[i][j] = s[x2 + 1][y2 + 1] - s[x1][y2 + 1] - s[x2 + 1][y1] + s[x1][y1];
            }
        }
        return ans;
    }
};

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func matrixBlockSum(mat [][]int, k int) [][]int {
    m, n := len(mat), len(mat[0])
    s := make([][]int, m+1)
    for i := range s {
        s[i] = make([]int, n+1)
    }
    for i, row := range mat {
        for j, x := range row {
            s[i+1][j+1] = s[i][j+1] + s[i+1][j] - s[i][j] + x
        }
    }

    ans := make([][]int, m)
    for i := range ans {
        ans[i] = make([]int, n)
    }

    for i := 0; i < m; i++ {
        for j := 0; j < n; j++ {
            x1 := max(i-k, 0)
            y1 := max(j-k, 0)
            x2 := min(m-1, i+k)
            y2 := min(n-1, j+k)
            ans[i][j] = s[x2+1][y2+1] - s[x1][y2+1] - s[x2+1][y1] + s[x1][y1]
        }
    }

    return ans
}

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function matrixBlockSum(mat: number[][], k: number): number[][] {
    const m: number = mat.length;
    const n: number = mat[0].length;

    const s: number[][] = Array.from({ length: m + 1 }, () => Array(n + 1).fill(0));
    for (let i = 0; i < m; i++) {
        for (let j = 0; j < n; j++) {
            s[i + 1][j + 1] = s[i][j + 1] + s[i + 1][j] - s[i][j] + mat[i][j];
        }
    }

    const ans: number[][] = Array.from({ length: m }, () => Array(n).fill(0));
    for (let i = 0; i < m; i++) {
        for (let j = 0; j < n; j++) {
            const x1: number = Math.max(i - k, 0);
            const y1: number = Math.max(j - k, 0);
            const x2: number = Math.min(m - 1, i + k);
            const y2: number = Math.min(n - 1, j + k);
            ans[i][j] = s[x2 + 1][y2 + 1] - s[x1][y2 + 1] - s[x2 + 1][y1] + s[x1][y1];
        }
    }

    return ans;
}

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